We have the second quantized representation \(U_{\mathrm {r}}:= \Gamma (U_1)\)  and the Fock vacuum
vector \(\Omega _{\mathrm {r}}\in \mathcal {H}_{\mathrm {r}}\) . This triple \((\mathcal {A}_{\mathrm {r}},U_{\mathrm {r}},\Omega _{\mathrm {r}})\)  is a two-dimensional
Haag-Kastler net and referred to as the free massive scalar net.
The subscript r is intended for the real scalar field.
Furthermore, this net satisfies the modular nuclearity [1]}, a property which we will
explain in more detail in Section .
